Kronenthal–Lazebnik's uniqueness conjecture for generalized quadrangles over algebraically closed fields

Let $\mathbb F$ be an algebraically closed field of characteristic zero, and let $f_2,f_3\in\mathbb F[p_1,l_1]$. Kronenthal–Lazebnik's conjecture. Every graph $B\Gamma_3(\mathbb F;f_2,f_3)$ with girth at least eight is isomorphic to

$$B\Gamma_3(\mathbb F;p_1l_1,p_1l_1^2).$$

The claim would establish uniqueness of this generalized-quadrangle construction over algebraically closed fields; the source presents it as expected and gives no resolution.